Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

Abstract

We investigate the behaviour of solution uu(x, t; λ) at λ =  λ* for the non-local porous medium equation \({u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}\) with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all \({x\in(-1,1)}\).

References

  1. 1

    Bebernes, J.W., Eberly, D.: Mathematical problems from combustion theory. Appl. Math. Sci. 83. Springer, Berlin (1989)

  2. 2

    Bebernes J.W., Lacey A.A.: Global existence and finite–time blow–up for a class of non-local parabolic problems. Adv. Differ. Equ. 2, 927–953 (1997)

    MATH  MathSciNet  Google Scholar 

  3. 3

    Bebernes J.W., Talaga P.: Non-local problems modelling shear banding. Commun. Appl. Nonlinear Anal. 3, 79–103 (1996)

    MATH  MathSciNet  Google Scholar 

  4. 4

    Bebernes J.W., Li C., Talaga P.: Single-point blow-up for non-local parabolic problems. Physica D 134, 48–60 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5

    Caglioti E., Lions P.-L., Marchioro C., Pulvirenti M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6

    Kavallaris N.I., Lacey A.A., Tzanetis D.E.: Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process. Nonlinear Anal. TMA 58, 787–812 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  7. 7

    Krzywicki A., Nadzieja T.: Some results concerning the Poisson–Boltzmann equation. Zastosowania Mat. (Appl. Math. (Warsaw)) 21, 265–272 (1991)

    MATH  MathSciNet  Google Scholar 

  8. 8

    Lacey A.A.: Thermal runaway in a non-local problem modelling Ohmic heating. Part I: Model derivation and some specail cases. Eur. J. Appl. Math. 6, 127–144 (1995)

    MATH  MathSciNet  Google Scholar 

  9. 9

    Lacey A.A.: Thermal runaway in a non–local problem modelling Ohmic heating. Part II: General proof of blow-up and asymptotics of runaway. Eur. J. Appl. Math. 6, 201–224 (1995)

    MATH  MathSciNet  Google Scholar 

  10. 10

    Latos, E.A., Tzanetis, D.E.: Existence and blow-up of solutions for a non-local filtration and porous medium problem. In: Proceedings of the Edinburgh Mathmatical Society (2009, in press)

  11. 11

    Liu Q., Liang F., Li Y.: Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process. Eur. J. Appl. Math. 20(3), 247–267 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  12. 12

    Ockendon J., Howison S., Lacey A., Movchan A.: Applied Partial Differential Equations. Oxford Univercity Press, Oxford (1999)

    Google Scholar 

  13. 13

    Sattinger D.H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21, 979–1000 (1972)

    MATH  Article  MathSciNet  Google Scholar 

  14. 14

    Tzanetis D.E.: Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating. Electron. J. Differ. Equ. 11, 1–26 (2002)

    MathSciNet  Google Scholar 

  15. 15

    Tzanetis D.E., Vlamos P.M.: A nonlocal problem modelling Ohmic heating with variable thermal conductivity. Nonlinear Anal. RWA 2, 443–454 (2001)

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Dimitrios E. Tzanetis.

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E. A. Latos was supported by the Greek State Scholarship Foundation (I.K.Y.).

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Latos, E.A., Tzanetis, D.E. Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source. Nonlinear Differ. Equ. Appl. 17, 137–151 (2010). https://doi.org/10.1007/s00030-009-0044-7

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Mathematics Subject Classification (2000)

  • Primary 35K55
  • Secondary 35B05

Keywords

  • Non-local parabolic problems
  • Porous medium
  • Grow-up of solutions